I can't find any other way than to just decompose the whole thing into sinA or cosA, and that is really long and usually full of mistakes. Is there something that i'm missing here? Something that makes this proof a lot easier?

Well, yeah, somewhat easier. Expand the right side and cancel the sin(2A) on both sides. Now expand sin(5A)=sin(2A+3A). Then maybe use sin(3A)cos(2A)-cos(3A)sin(2A)=sin(A)? It goes a bit easier, yes? Hope I don't have a typo in there.